Introduction to Limits (DP IB Analysis & Approaches (AA)): Revision Note

Limits

What are limits in mathematics?

• A limit in mathematics involves the tendency of a mathematical process as it approaches, but never quite reaches, an ‘end point’ of some sort

• We use a special limit notation to indicate this

  • For example  denotes ‘the limit of the function f(x) as x goes to (or approaches) 3’

    • I.e., what value (if any) does f(x) get closer and closer to as x takes on values closer and closer to 3

    • We are not concerned here with what value (if any) f(x) takes when x is equal to 3

      • only with the behaviour of f(x) as x gets close to 3

• The sum of an infinite geometric sequence is a type of limit

  • When you calculate for an infinite geometric sequence, you are actually finding 

    • I.e., what value (if any) the sum of the first n terms of the sequence gets closer and closer to as the number of terms (n) goes to infinity

    • The sum never actually reaches

      • but as more and more terms are included in the sum it gets closer and closer to that value

What can I be asked to do with limits?

• In the IB course you will normally be considering the limits of functions

  • This may include finding the limit at a point where the function is undefined

    • For example,  is undefined when x = 0

      • but you might want to know how the function behaves as x gets closer and closer to zero

  • Or it may include finding the limit of a function f(x) as x gets infinitely big in the positive or negative direction

    • For this type of limit we write  or  

      • (The first one can also be written as  to distinguish it from the second one)

    • These sorts of limits are often used to find the asymptotes of the graph of a function

How do I find a simple limit?

• STEP 1
  To find  begin by substituting a into the function f(x)

  • If f(a) exists with a well-defined value, then that is also the value of the limit

    • E.g., for

      • 

    • In this case,  is simply equal to f(3)

• STEP 2
  If f(a) does not exist, it may be possible to simplify f(x) so that substituting a into the simplified function gives a well-defined value

  • In that case, the value of the simplified version of the function is also the value of the limit of the function as x goes to a

    • E.g.  is not defined at x = 0, but you may use algebra to find the limit as x approaches zero

      • 

    • Note that  and  are not the same function!

      • They are equal for all values of x except zero

      • But for x = 0, g(0) = 0 while f(0) is undefined

      • However f(x) gets closer and closer to zero as x gets closer and closer to zero

• If neither of these steps gives a well-defined value for the limit you may need to consider more advanced techniques to evaluate the limit

  • For example l’Hôpital’s Rule or using Maclaurin series

How do I find a limit to infinity?

• As x goes to  or , a function f(x) may converge to a well-defined value, or it may diverge to or 

  • Other behaviours are possible

    • For example  is simply undefined, because sin x continues to oscillate between 1 and -1 as x gets larger and larger

• There are two key results to be used here:

  •   converges to 0 for all n >0 and all 

  •  diverges to  for all n > 0

    •  for n > 0 will need to be considered on a case-by-case basis

      • xⁿ behaves differently for different values of n when x is negative

• STEP 1
  If necessary, use algebra to rearrange the function into a form where one or the other of the key results above may be applied

• STEP 2
  Use the key results above to evaluate your limit
   

• For example

  • 

• Or

  • 
    

    • I.e., the limit diverges to (because  gets bigger and bigger without limit as x gets bigger and bigger)

• Another useful result is that if , then for any 

  • This can be useful for example when evaluating the limits of functions containing exponentials

    • E.g.  for any p > 0,

      • So you immediately know  for p > 0

    • See the worked example below for a more involved version of this

Do limits ever have ‘directions’?

• Yes they do!

• The notation means ‘the limit of f(x) as x approaches a from above’

  • I.e., this is the limit as x comes ‘down’ towards a

    • It only considers the function’s behaviour for values of x that are greater than a

• The notation means ‘the limit of f(x) as x approaches a from below’

  • I.e., this is the limit as x comes ‘up’ towards a

    • It only considers the function’s behaviour for values of x that are less than a

• One place these sorts of limits appear is for functions defined piecewise

  • Limits ‘from above’ and ‘from below’ may well be different at values of x where the different ‘pieces’ of the function are joined

• But also be aware of a situation like the following:

  •  

    • Because  for x > 0, with  becoming bigger and bigger in the positive direction as x gets closer and closer to zero ‘from above’

  • 

    • Because  for x < 0, with  becoming bigger and bigger in the negative direction as x gets closer and closer to zero ‘from below’

  • The graph of  shows this limiting behaviour as x approaches zero from the two different directions